integral of 1/(sqrt((x-1)(3-x)))

解答

\int \frac{1}{( x - 1 ) ( 3 - x )} d x

arcsin (x - 2) + C

求解步骤

\int \frac{1}{( x - 1 ) ( 3 - x )} d x

对

(x - 1) (3 - x)

配方:

- (x - 2)^{2} + 1

= \int \frac{1}{- ( x - 2 ) ^{2} + 1} d x

使用换元积分法

= \int \frac{1}{- u ^{2} + 1} d u

使用积分常见定则:

\int \frac{1}{- u ^{2} + 1} d u = arcsin (u)

= arcsin (u)

u = x - 2

代回

= arcsin (x - 2)

解答补常数

= arcsin (x - 2) + C

\int (\frac{1}{1 + y ^{2}}) d y

\int \frac{4 x ^{3}}{x ^{2} - 16} d x

\int \frac{25}{x ^{2} - 25} d x

\int \frac{1}{( x ^{2} - 1 )} d x

\int 1 - 3 x d x